arXiv subjects

Liangyun Zhang

Publications and source records attributed to Liangyun Zhang.

8 recordsLinked to original sources

arXiv2026

Hopf heap modules, Rota-Baxter operators, and their structure theorems

This paper is primarily devoted to the study of Hopf heaps and Hopf heap modules. We redefine the structure of Hopf trusses by means of Hopf heaps, establish the connection between Hopf trusses and Hopf braces, and provide a series of examples of Hopf truss structures from the perspective of Hopf heaps. Most importantly, we introduce the conception of Hopf heap modules, and present its structure theorem. Finally, we introduce the notions of Rota-Baxter operators on Hopf heaps and Hopf heap modules, and present the structure theorem for Rota-Baxter Hopf heap modules.

math.RA↗

arXiv2024

Rota-Baxter operators on cocommutative Hopf algebras and Hopf braces

This paper studies the relationship of Rota-Baxter operators on cocommutative Hopf algebras with Hopf braces and the Yang-Baxter equation, with emphasis on the embedding of cocommutative Hopf braces into Rota-Baxter Hopf algebras. Through Hopf braces, we establish a connection between relative Rota-Baxter operators on cocommutative Hopf algebras and bijective 1-cocycles. Finally, we introduce the notion of symmetric Hopf braces, and establish the relationship between symmetric Hopf braces and Rota-Baxter Hopf algebras.

math.QA↗

arXiv2020

Hopf-Galois algebras and their Poisson structures

As is known to all, Hopf-Galois objects have a significant research value for analyzing tensor categories of comodules and classification questions of pointed Hopf algebras, and are natural generalizations of Hopf algebras with a Galois-theoretic flavour. In this paper, we mainly prove a criterion for an Ore extension of a Hopf-Galois algebra to be a Hopf-Galois algebra, and introduce the conception of Poisson Hopf-Galois algebras, and establish the relationship between Poisson Hopf-Galois algebras and Poisson Hopf algebras. Moreover, we study Poisson Hopf-Galois structures on Poisson polynomial algebras, and mainly give a necessary and sufficient condition for the Poisson enveloping algebra of a Poisson Hopf-Galois algebra to be a Hopf-Galois algebra.

math.QA↗

arXiv2019

Hopf brace, braid equation and bicrossed coproduct

In this paper, we mainly give some equivalent characterisations of Hopf braces, show that the category $\mathcal{CB}(A)$ of Hopf braces is equivalent to the category $\mathcal{C}(A)$ of bijective 1-cocycles, and prove that the category $\mathcal{CB}(A)$ of Hopf braces is also equivalent to the category $\mathcal{M}(A)$ of Hopf matched pairs. Moreover, we construct many more Hopf braces on polynomial Hopf algebras, Long copaired Hopf algebras and Drinfel'd doubles of finite dimensional Hopf algebras, and give a sufficient and necessary condition for a given bicrossed coproduct $A\bowtie H$ to be a Hopf brace if $A$ or $H$ is a Hopf brace.

math.RA↗

arXiv2017

Rota-Baxter paired modules and their constructions from Hopf algebras

In this paper, we introduce the concept of a Rota-Baxter paired module to study Rota-Baxter modules without necessarily a Rota-Baxter operator. We obtain two characterizations of Rota-Baxter paired modules, and give some basic properties of Rota-Baxter paired modules. Beginning with the connection between the notion of integrals in the representations of Hopf algebra and of the notion of an integral algebra as a motivation and special case of Rota-Baxter algebra of weight zero, we obtain a large number of Rota-Baxter paired modules from Hopf related algebras and modules, including semisimple Hopf algebras, weak Hopf algebras, Long bialgebras, quasitriangular Hopf algebras, weak Hopf modules, dimodules and Doi-Hopf modules. Some of them give new examples of Rota-Baxter algebras.

math.QA↗

arXiv2017

Non-trivially graded self-dual fusion categories of rank $4$

Let $\mathcal{C}$ be a self-dual spherical fusion categories of rank $4$ with non-trivial grading. We complete the classification of Grothendieck ring $K(\mathcal{C})$ of $\mathcal{C}$; that is, we prove that $K(\mathcal{C})\cong Fib\otimes\mathbb{Z}[\mathbb{Z}_2]$, where $Fib$ is the Fibonacci fusion ring and $\mathbb{Z}[\mathbb{Z}_2]$ is the group ring on $\mathbb{Z}_2$. In particular, if $\mathcal{C}$ is braided then it is equivalent to $\textbf{Fib}\boxtimes\textbf{Vec}_{\mathbb{Z}_2}^ω$ as fusion categories, where $\textbf{Fib}$ is a Fibonacci category and $\textbf{Vec}_{\mathbb{Z}_2}^ω$ is a rank $2$ pointed fusion category.

math.RA↗

arXiv2014

Hopf-Galois extensions for monoidal Hom-Hopf algebras

We investigate the theory of Hopf-Galois extensions for monoidal Hom-Hopf algebras. As the main result of this paper, we prove the Schneider's affineness theorems in the case of monoidal Hom-Hopf algebras in terms of the theory of the total integral and Hom-Hopf Galois extensions. In addition, we obtain the affineness criterion for relative Hom-Hopf module associated with faithfully flat Hom-Hopf Galois extensions.

math.RA↗

arXiv2013

On Hopf monoids in duoidal categories

Aguiar and Mahajan's bimonoids A in a duoidal category M are studied. Under certain assumptions on M, the Fundamental Theorem of Hopf Modules is shown to hold for A if and only if the unit of A determines an A-Galois extension. Our findings are applied to the particular examples of small groupoids and of Hopf algebroids over a commutative base algebra.

math.QA↗